https://github.com/colgreen/Redzen/blob/master/Redzen/Random...
There's an explanation in the comments, and a link to the original fuller explanation and source, here:
https://mumble.net/~campbell/tmp/random_real.c
I.e. I'm pretty sure your approach gives a non uniform distribution, but happy to discuss in more detail if you want.
Great sources!
(EDIT: replace "omit" by "leave out", because it's too close to "emit")
Unless I made some significant error, it is a uniform distribution over multiples of 2^-52. It's also a discrete approximation of the real uniform distribution in that for any subdivision of [0,1) into N equal (half-open) intervals, the most-populated interval has at most one more inhabitant the least populated one.
It is however strictly inferior to rand53()*2^-53 (NextDoubleInner at your github link) unless your int64->double or float-multiply operations are quite bad.